Collective viscous propulsion of a two-dimensional flotilla of Marangoni boats

Crowdy, Darren

doi:10.1103/physrevfluids.5.124004

Cited by 15 publications

(22 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The present paper is a sequel to the author's previous study 17 and shows how to incorporate reaction effects into that two-dimensional Marangoni boat model. An understanding of reaction effects is important because the latter lead to a removal of surfactants from the interface.…”

Section: Introductionmentioning

confidence: 93%

“…For a flat interface on y = 0 it is natural to view the surface tension distribution as a function of x, σ = σ(x). It is a common model to take a linear equation of state 16,17,21,22 relating the surfactant concentration Γ(x) to the local surface tension value:…”

Section: Solution For the Surfactant Distributionmentioning

confidence: 99%

“…Under the same assumptions on the Reynolds, capillary and surface Péclet numbers Masoud and Stone 31 used the reciprocal theorem for Stokes flow to calculate the propulsion speed of active oblate and prolate spheroids suspended at interfaces. And motivated by the observation of collective "uniform flow" states of multiple boats interacting on a free surface 18,19 a recent study 17 finds an analytical solution for the Marangoni-induced flow of a periodic array, or "flotilla", of identical boats driven purely by surface diffusion of insoluble surfactant in a two-dimensional setting. Even in the simpler two-dimensional setting, and with the same assumptions of zero Reynolds, capillary and surface Péclet number used by other authors 16,31,36 , the relevant problems that incorporate Marangoni effects are challenging mixed boundary value problems requiring proper resolution of the associated contact line stress singularities.…”

Section: Introductionmentioning

confidence: 99%

“…The methods of complex analysis can, however, be deployed to great advantage in such two-dimensional problems. For a flotilla of two-dimensional Marangoni boats such methods were used to derive an explicit expression for its collective propulsion speed, as well as an analytical description of the low-Reynoldsnumber Marangoni-induced flow beneath it 17 . As discussed in that study 17 the solution for propulsion of a single two-dimensional boat driven purely by surface diffusion does not exist, a matter discussed in more detail later.…”

Section: Introductionmentioning

confidence: 99%

“…In an experimental study of vapour-driven propulsion on a free surface 20 steady propulsion of a floating boat is achieved through a continuous supply of fuel vapour that lowers the surface tension of the liquid coupled with a spontaneous recovery of the surface tension after the boat has passed caused by evaporation of the surfactant off the interface. To study such a scenario we extend the model of diffusion-driven motion of a flotilla of Marangoni boats 17 . Here, however, we restrict attention to a single boat but incorporate the additional physical effect of first-order reaction kinetics; this regularizes the two-dimensional problem making a solution possible for a single boat.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

Crowdy

2021

Phys. Rev. Fluids

Self Cite

View full text Add to dashboard Cite

An analytical solution is derived for the flow generated by a self-propelling twodimensional Marangoni boat driven by reactive insoluble surfactant on a deep layer of fluid of viscosity µ at zero Reynolds number, capillary number and surface Péclet number. In the model, surfactant emitted from the edges of the boat causes a surface tension disparity across the boat. Once emitted, the surfactant diffuses along the interface and sublimates to the upper gas phase. A linear equation of state relates the surface tension to the surfactant concentration. The propulsion speed of the boat is shown to bewhere Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, ∆σ is the surface tension disparity between the front and rear of the boat and K 0 is the order-zero modified Bessel function. Explicit expressions for the streamfunction associated with the Stokes flow beneath the boat are found facilitating ready examination of the Marangoni-induced streamlines. An integral formula, derived using the reciprocal theorem, is also given for the propulsion speed of the boat in response to a more general Marangoni stress distribution.

show abstract

Section: Introductionmentioning

confidence: 93%

Section: Solution For the Surfactant Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

Crowdy

2021

Phys. Rev. Fluids

Self Cite

View full text Add to dashboard Cite

show abstract

Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

Crowdy

2021

J Eng Math

View full text Add to dashboard Cite

A class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

show abstract

Cole–Hopf linearization of the thermocapillary Marangoni dynamics of a two-dimensional bubble with insoluble surfactant

Crowdy

2024

J Eng Math

View full text Add to dashboard Cite

The Marangoni stress-induced dynamics of a two-dimensional inviscid bubble loaded with insoluble surfactant moving in a linear temperature gradient is studied in the limit of small Reynolds, capillary and thermal Péclet numbers. The bubble moves due to the combined effects of thermocapillary Marangoni stresses and those induced by the advective-diffusive spreading of an initial concentration of surfactant on the bubble boundary. It is shown that this nonlinear multiphysics initial value problem is linearizable at any finite non-zero value of a surface Péclet number $$Pe_s$$ P e s governing the size of surface diffusion of the insoluble surfactant relative to its advective spreading. This is done by showing that the dynamics can be encoded in the evolution of a function, analytic and single-valued outside the bubble, that satisfies a complex partial differential equation of Burgers type. It is shown that this equation can be linearized by a complex generalization of the classical Cole–Hopf transformation. A numerical method is formulated to solve this linear partial differential equation and determine the Marangoni dynamics of a bubble with some initial surfactant concentration and illustrative calculations are carried out. Results are shown to be consistent with exact equilibrium solutions available from the formulation as $$Pe_s \rightarrow 0$$ P e s → 0 and $$Pe_s \rightarrow \infty $$ P e s → ∞ and a perturbative formula for the bubble migration velocity at large but finite $$Pe_s$$ P e s .

show abstract

Collective viscous propulsion of a two-dimensional flotilla of Marangoni boats

Cited by 15 publications

References 36 publications

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

Cole–Hopf linearization of the thermocapillary Marangoni dynamics of a two-dimensional bubble with insoluble surfactant

Contact Info

Product

Resources

About